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speeding up matrix multiplication (newbie question)
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Date: 2009-02-20 (22:31)
From: Will M. Farr <farr@m...>
Subject: Re: [Caml-list] speeding up matrix multiplication (newbie question)
Mike and Erick,

In some of my work, I've got code which is constantly creating and
multiplying 4x4 matrices (Lorentz transforms).  I usually write in a
functional style, so I do not generally overwrite old matrices with
the multiplication results.  I have discovered that, at these sizes,
it's about a factor of two to three faster to do this with OCaml
arrays for the matrices rather than the Bigarrays you would need to
use to interface with the BLAS in Lacaml or ocamlgsl.  The reason is
that the memory block for a bigarray is malloc-ed, which is extremely
slow compared to the OCaml native allocation.  That overhead
completely eliminates the advantage of going through C or Fortran

The ideal solution would be to use OCaml native float arrays (fast
allocation) with LAPACK (fast matrix multiply), but I'm satisfied with
the performance of my code, and too lazy to put out the effort to
implement the C glue for this.

Just thought you might appreciate another data point.  I have no idea
where the turnover is between 4x4 and 61x61 :).


2009/2/20 Mike Lin <nilekim@gmail.com>:
> Erick, we should compare notes sometime. I have a lot of code for doing this
> kind of stuff (I am working on empirical codon models with 61x61 rate
> matrices). The right way to speed up matrix-vector operations is to use BLAS
> via either Lacaml or ocamlgsl. But if, like me, you like to
> counterproductively fiddle around with stupid things, here's a little ocaml
> library I wrote that links to a C function to do vector-vector dot using
> SSE2 vectorization.
> http://www.broad.mit.edu/~mlin/for_others/SuperFast_dot.zip
> IIRC in my tests it gave about 50% speedup vs. the obvious C code and 100%
> speedup vs. the obvious ocaml code. But, I am doing 61x61 and I'm not sure
> if the speedup scales down to 20x20 or especially 4x4.
> Mike
> On Fri, Feb 20, 2009 at 2:53 PM, Erick Matsen <matsen@berkeley.edu> wrote:
>> Wow, once again I am amazed by the vitality of this list. Thank you
>> for your suggestions.
>> Here is the context: we are interested in calculating the likelihood
>> of taxonomic placement of short "metagenomics" sequence fragments from
>> unknown organisms in the ocean. We start by assuming a model of
>> sequence evolution, which is a reversible Markov chain. The taxonomy
>> is represented as a tree, and the sequence information is a collection
>> of likelihoods of sequence identities. As we move up the tree, these
>> sequences "evolve" by getting multiplied by the exponentiated
>> instantaneous Markov matrix.
>> The matrices are of the size of the sequence model: 4x4 when looking
>> at nucleotides, and 20x20 when looking at proteins.
>> The bottleneck is (I mis-spoke before) that we are multiplying many
>> length-4 or length-20 vectors by a collection of matrices which
>> represent the time evolution of those sequences as follows.
>> Outer loop:
>>  modify the amount of time each markov process runs
>>  exponentiate the rate matrices to get transition matrices
>>  Recur over the tree, starting at the leaves:
>>    at a node, multiply all of the daughter likelihood vectors together
>>    return the multiplication of that product by the trasition matrix
>> (bottleneck!)
>> The trees are on the order of 50 leaves, and there are about 500
>> likelihood vectors done at once.
>> All of this gets run on a big cluster of Xeons. It's not worth
>> parallelizing because we are running many instances of this process
>> already, which fills up the cluster nodes.
>> So far I have been doing the simplest thing possible, which is just to
>> multiply the matrices out like \sum_j a_ij v_j. Um, this is a bit
>> embarassing.
>> let mul_vec m v =
>>    if Array.length v <> n_cols m then
>>      failwith "mul_vec: matrix size and vector size don't match!";
>>    let result = Array.create (n_rows m) N.zero in
>>    for i=0 to (n_rows m)-1 do
>>      for j=0 to (n_cols m)-1 do
>>        result.(i) <- N.add result.(i) (N.mul (get m i j) v.(j))
>>      done;
>>    done;
>>    result
>> I have implemented it in a functorial way for flexibility. N is the
>> number class. How much improvement might I hope for if I make a
>> dedicated float vector multiplication function? I'm sorry, I know
>> nothing about "boxing." Where can I read about that?
>> Thank you again,
>> Erick
>> On Fri, Feb 20, 2009 at 10:46 AM, Xavier Leroy <Xavier.Leroy@inria.fr>
>> wrote:
>> >> I'm working on speeding up some code, and I wanted to check with
>> >> someone before implementation.
>> >>
>> >> As you can see below, the code primarily spends its time multiplying
>> >> relatively small matrices. Precision is of course important but not
>> >> an incredibly crucial issue, as the most important thing is relative
>> >> comparison between things which *should* be pretty different.
>> >
>> > You need to post your matrix multiplication code so that the regulars
>> > on this list can tear it to pieces :-)
>> >
>> > From the profile you gave, it looks like you parameterized your matrix
>> > multiplication code over the + and * operations over matrix elements.
>> > This is good for genericity but not so good for performance, as it
>> > will result in more boxing (heap allocation) of floating-point values.
>> > The first thing you should try is write a version of matrix
>> > multiplication that is specialized for type "float".
>> >
>> > Then, there are several ways to write the textbook matrix
>> > multiplication algorithm, some of which perform less boxing than
>> > others.  Again, post your code and we'll let you know.
>> >
>> >> Currently I'm just using native (double-precision) ocaml floats and
>> >> the native ocaml arrays for a first pass on the problem.  Now I'm
>> >> thinking about moving to using float32 bigarrays, and I'm hoping
>> >> that the code will double in speed. I'd like to know: is that
>> >> realistic? Any other suggestions?
>> >
>> > It won't double in speed: arithmetic operations will take exactly the
>> > same time in single or double precision.  What single-precision
>> > bigarrays buy you is halving the memory footprint of your matrices.
>> > That could result in better cache behavior and therefore slightly
>> > better speed, but it depends very much on the sizes and number of your
>> > matrices.
>> >
>> > - Xavier Leroy
>> >
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